Conformal Universality in Normal Matrix Ensembles

TL;DR

This paper investigates the universality of eigenvalue statistics in random normal matrix ensembles, revealing that conformal transformations can induce universality where it does not naturally occur, unlike in Hermitian ensembles.

Contribution

It demonstrates that universality in normal matrix ensembles requires conformal transformations, contrasting with Hermitian ensembles where universality is inherent.

Findings

Eigenvalue statistics of normal matrices can be made universal via conformal maps.

Universality concepts from Hermitian ensembles do not directly apply to normal ensembles.

Normal matrix ensembles exhibit unique universality properties under specific transformations.

Abstract

A remarkable property of Hermitian ensembles is their universal behavior, that is, once properly rescaled the eigenvalue statistics does not depend on particularities of the ensemble. Recently, normal matrix ensembles have attracted increasing attention, however, questions on universality for these ensembles still remain under debate. We analyze the universality properties of random normal ensembles. We show that the concept of universality used for Hermitian ensembles cannot be directly extrapolated to normal ensembles. Moreover, we show that the eigenvalue statistics of random normal matrices with radially symmetric potential can be made universal under a conformal transformation.